group of units

Theorem.

Proof. If u and v are two units, then there are the elements r and s of R such that r⁢u=u⁢r=1 and s⁢v=v⁢s=1. Then we get that (s⁢r)⁢(u⁢v)=s⁢(r⁢(u⁢v))=s⁢((r⁢u)⁢v)=s⁢(1⁢v)=s⁢v=1, similarly (u⁢v)⁢(s⁢r)=1. Thus also u⁢v is a unit, which means that E is closed under multiplication. Because 1∈E and along with u also its inverser belongs to E, the set E is a group.